Back to Graph Theory

## Rules

Chomp is played on a rectangular grid, such as squares of a candy bar. The lower left square is considered "poison". Players take turns picking a square. With each choice, all squares above and to the right of the picked square are no longer available --they are eaten. The person forced to take the "poison" square loses.

Example: Playing on a 3x8 grid, the lower left square (in black) is the poison square. The first player chooses the red square of the grid and all the blue squares are eaten.

Then the second player chooses the yellow square:

The first player responds with the red square:

The second player plays the yellow square:

The first player plays the red square:

Now the second player must choose the black poison square and loses.

## Questions

Use a piece of graph paper and try playing some games on a variety of sizes of boards. Below are a few questions to consider to help you think about strategies to win.

1. Who has a winning strategy? Is either player able to control the game? If so, which player can assure he/she wins?
2. Winning on a square board: Consider square boards (3x3, 4x4, 5x5, ...) Can you find any positions from which you can guarantee you can win?
3. Winning on a 2xn board: Consider boards with 2 rows (2x3, 2x4, 2x5,...) Can you find any positions from which you can guarantee you can win? How can you make sure that you can get to those positions?

NOTE: The 3xn case was just solved in 2002 by a high school senior, Steven Byrnes of Lexington, MA, as part of a larger theorem. He won the Siemens-Westinghouse Competition in Math, Science and Technology.